Optimal. Leaf size=202 \[ \frac{148780 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt{3 x^2+5 x+2}}-\frac{59512}{81} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}-\frac{110516 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.128699, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {818, 832, 843, 718, 424, 419} \[ -\frac{2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt{3 x^2+5 x+2}}-\frac{59512}{81} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}+\frac{148780 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{110516 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
Rule 832
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2}{9} \int \frac{(3+2 x)^{5/2} (4+411 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt{2+5 x+3 x^2}}+\frac{4}{27} \int \frac{(-18243-22317 x) \sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt{2+5 x+3 x^2}}-\frac{59512}{81} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{8}{243} \int \frac{-34269-\frac{82887 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt{2+5 x+3 x^2}}-\frac{59512}{81} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{55258}{81} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{74390}{81} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt{2+5 x+3 x^2}}-\frac{59512}{81} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{\left (110516 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{81 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (148780 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{81 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt{2+5 x+3 x^2}}-\frac{59512}{81} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{110516 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{148780 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.500917, size = 220, normalized size = 1.09 \[ -\frac{2 \left (2 \left (3 x^2+5 x+2\right ) \left (-5312 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+55258 \left (3 x^2+5 x+2\right )+27629 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )+3 (2 x+3) \left (144 x^4-166566 x^3-411640 x^2-330053 x-85285\right )\right )}{243 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 325, normalized size = 1.6 \begin{align*}{\frac{2}{ \left ( 2430+3645\,x \right ) \left ( 6\,{x}^{2}+13\,x+6 \right ) \left ( 1+x \right ) ^{2}}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 28698\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+82887\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+47830\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+138145\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+19132\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +55258\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -4320\,{x}^{5}+4990500\,{x}^{4}+19844670\,{x}^{3}+28425390\,{x}^{2}+17410935\,x+3837825 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (2 \, x + 3\right )}^{\frac{9}{2}}{\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (16 \, x^{5} + 16 \, x^{4} - 264 \, x^{3} - 864 \, x^{2} - 999 \, x - 405\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (2 \, x + 3\right )}^{\frac{9}{2}}{\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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